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Elementary Functional Analysis (Dover Books on Mathematics), by Georgi E. Shilov
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In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions (hence the term "functional analysis"), rather than sets of points in a finite-dimensional space.
Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space. The solution of the linear equation with constant (operator) coefficients is written in general form in terms of the exponential of the operator. This leads, in the finite-dimensional case, to explicit formulas not only for the solutions of first-order equations, but also to the solutions of higher-order equations and systems of equations. The third chapter presents a theory of curvature for curve in a multidimensional space.
The final two chapters essentially comprise an introduction to Fourier analysis. In the treatment of orthogonal expansions, a key role is played by Fourier series and the various kinds of convergence and summability for such series. The material on Fourier transforms, in addition to presenting the more familiar theory, also deals with problems in the complex domain, in particular with problems involving the Laplace transform.
Designed for students at the upper-undergraduate or graduate level, the text includes a set of problems for each chapter, with hints and answers at the end of the book.
- Sales Rank: #1262201 in Books
- Published on: 1996-01-18
- Released on: 1996-01-18
- Original language: English
- Number of items: 1
- Dimensions: 8.44" h x .67" w x 5.37" l, .70 pounds
- Binding: Paperback
- 352 pages
Language Notes
Text: English (translation)
Original Language: Russian
About the Author
Soviet mathematician Georgi E.�Shilov (1917-75) was a longtime Professor of Mathematics at Moscow State University and�the author of several other successful Dover reprints.
Most helpful customer reviews
38 of 41 people found the following review helpful.
A beginning course in functional analysis
By Palle E T Jorgensen
Elementary Functional Analysis by Georgi E. Shilov is suitable for a beginning course in functional analysis and some of its applications, e.g., to Fourier series, to harmonic analysis, to partial differential equations (PDEs), to Sobolev spaces, and it is a good supplement and complement to two other popular books in the subject, one by Rudin, and another by Edwards.
Rudin's book is entitled "Functional Analysis" (not in the Dover series) and it is my favorite. Rudin's book is of newer vintage, and it goes more in depth, and includes new material on unbounded operators in Hilbert space. Edwards' book "Functional Analysis: Theory and Applications;" is in the Dover series, and it is twice as thick as Shilov's book.
Topics covered in Shilov: Function spaces, L^p-spaces, Hilbert spaces, and linear operators; the standard Banach, and Hahn-Banach theorems. It includes many exercises and examples. Well motivated with applications.
Book Comparison:
Shilov book is gentler on students, and it is probably easier to get started with: It stresses motivation a bit more, the exercises are easier, and finally Shilov includes a few applications; fashionable these days. And of course, the books in the Dover series are cheap in comparison. Review by Palle Jorgensen, August 2007.
15 of 19 people found the following review helpful.
The book is very good but it is not self contained
By R. Cardoso
I enjoyed the last chapters of the book, that are relative to the applications: differential geometry, Fourier series, Fourier transform, differential equations.
Also I think that the study of the convergence of Fourier series partially based on the notion of delta-like sequences of functions is very interesting and possibly it enables some generalizations.
However the book is not self-contained because it does very much references to 'Linear Algebra' and 'Elementary Real and Complex Analysis' of the author.
62 of 62 people found the following review helpful.
another great book from Shilov
By Jonathan Hay
This is volume 2 of Shilov's Moscow University course on Mathematical Analysis. If you have read Shilov's first volume ("Elementary Real and Complex Analysis") or his volume on Linear Algebra, then you can expect more of the same clear explanation and thoughtful organization of materials. His proofs are designed to help the reader understand material and provide deep insight into the mathematics involved. Highly recommended for those who want a consise -- but very thorough -- introduction to theory behind differential equations and Fourier analysis.
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